Second law of quantum complexity

We investigate the first law of complexity proposed in [1], i.e., the variation of complexity when the target state is perturbed, in more detail. Based on Nielsen's geometric approach to quantum circuit complexity, we find the variation only depends on the end of the optimal circuit. We apply the first law to gain new insights into the quantum circuits and complexity models underlying holographic complexity. In particular, we examine the variation of the holographic complexity for both the complexity=action and complexity=volume conjectures in perturbing the AdS vacuum with coherent state excitations of a free scalar field. We also examine the variations of circuit complexity produced by the same excitations for the free scalar field theory in a fixed AdS background. In this case, our work extends the existing treatment of Gaussian coherent states to properly include the time dependence of the complexity variation. We comment on the similarities and differences of the holographic and QFT results.

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“…satisfying the commutation relation [φ n ,π † n ] = iδ n n . 25 The Hamiltonian can be expressed in terms of the normal mode field operators aŝ…”

Section: Quantized Scalar Field In Ads D+1mentioning

confidence: 99%

Aspects of the first law of complexity

Bernamonti¹,

Galli²,

Hernandez³

et al. 2020

J. Phys. A: Math. Theor.

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“…Roughly speaking, the complexity of a pure quantum state is the minimal number of gates of any quantum circuit built from a fixed set of gates that produces this state from a given reference state. Complexity was first studied within the framework of the

AdS / CFT

correspondence in the context of time‐dependent thermal state complexity, which was proposed to be dual either to the volume of the Einstein‐Rosen bridge, or the action of a Wheeler‐DeWitt patch . Recently, additional insight has been gained into both proposals from more detailed holographic studies …”

Section: Introductionmentioning

confidence: 99%

“…Complexity was first studied within DOI: 10.1002/prop.201800034 the framework of the AdS/CFT correspondence in the context of timedependent thermal state complexity, which was proposed to be dual either to the volume of the Einstein-Rosen bridge, [9] or the action of a Wheeler-DeWitt patch. [10] Recently, additional insight has been gained into both proposals from more detailed holographic studies. [11][12][13] The present paper is concerned with the subregion complexity of the reduced density matrix of a finite subregion A.…”

Section: Introductionmentioning

confidence: 99%

Topological Complexity in AdS₃/CFT₂

Abt

Erdmenger

Hinrichsen

et al. 2018

Fortschritte der Physik

106

123

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We consider subregion complexity within the AdS3/CFT2 correspondence. We rewrite the volume proposal, according to which the complexity of a reduced density matrix is given by the spacetime volume contained inside the associated Ryu‐Takayanagi (RT) surface, in terms of an integral over the curvature. Using the Gauss‐Bonnet theorem we evaluate this quantity for general entangling regions and temperature. In particular, we find that the discontinuity that occurs under a change in the RT surface is given by a fixed topological contribution, independent of the temperature or details of the entangling region. We offer a definition and interpretation of subregion complexity in the context of tensor networks, and show numerically that it reproduces the qualitative features of the holographic computation in the case of a random tensor network using its relation to the Ising model. Finally, we give a prescription for computing subregion complexity directly in CFT using the kinematic space formalism, and use it to reproduce some of our explicit gravity results obtained at zero temperature. We thus obtain a concrete matching of results for subregion complexity between the gravity and tensor network approaches, as well as a CFT prescription.

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“…[27][28][29][30][31]) and the gravity action in Wheeler-De Witt patch in [32,33] (for recent progresses see e.g. [34][35][36][37][38][39][40][41][42]). We would also like to mention that for CFTs, the behavior of the complexity is very similar to the quantum information metric under marginal deformations as pointed out in [43] (refer to [27,31,44] for recent developments), where the metric is argued to be well approximated by the volume of maximal time slice in AdS.…”

Section: Jhep11(2017)097mentioning

confidence: 99%

Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT

Caputa

Kundu

Miyaji

et al. 2017

J. High Energ. Phys.

285

402

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Abstract:We propose an optimization procedure for Euclidean path-integrals that evaluate CFT wave functionals in arbitrary dimensions. The optimization is performed by minimizing certain functional, which can be interpreted as a measure of computational complexity, with respect to background metrics for the path-integrals. In two dimensional CFTs, this functional is given by the Liouville action. We also formulate the optimization for higher dimensional CFTs and, in various examples, find that the optimized hyperbolic metrics coincide with the time slices of expected gravity duals. Moreover, if we optimize a reduced density matrix, the geometry becomes two copies of the entanglement wedge and reproduces the holographic entanglement entropy. Our approach resembles a continuous tensor network renormalization and provides a concrete realization of the proposed interpretation of AdS/CFT as tensor networks. The present paper is an extended version of our earlier report arXiv:1703.00456 and includes many new results such as evaluations of complexity functionals, energy stress tensor, higher dimensional extensions and time evolutions of thermofield double states.

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Second law of quantum complexity

Cited by 281 publications

References 55 publications

Aspects of the first law of complexity

Aspects of the first law of complexity

Topological Complexity in AdS₃/CFT₂

Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT

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Second law of quantum complexity

Cited by 281 publications

References 55 publications

Aspects of the first law of complexity

Aspects of the first law of complexity

Topological Complexity in AdS3/CFT2

Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT

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Product

Resources

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Topological Complexity in AdS₃/CFT₂